Homotopy Type Theory and Algebraic Model Structures (I) Nicola - PowerPoint PPT Presentation

Homotopy Type Theory and Algebraic Model Structures (I) Nicola Gambino School of Mathematics University of Leeds Topologie Alg´ ebrique et Applications Paris, 2nd December 2016 1

Plan of the talks Goal ◮ analysis of the cubical set model of Homotopy Type Theory By-products ◮ general method to obtain right proper algebraic model structures, ◮ a new proof of model structure for Kan complexes and its right properness, avoiding minimal fibrations. 2

Outline of this talk Part I: Homotopy Type Theory ◮ Models of HoTT ◮ Quillen model structures ◮ Some issues Part II: Uniform fibrations and the Frobenius property ◮ Algebraic weak factorization systems ◮ Uniform fibrations ◮ The Frobenius property 3

References 1. C. Cohen, T. Coquand, S. Huber and A. M¨ ortberg Cubical Type Theory: a constructive interpretation of the univalence axiom arXiv, 2016. 2. N. Gambino and C. Sattler Frobenius condition, right properness, and uniform fibrations arXiv, 2016. 4

Part I: Homotopy Type Theory 5

Homotopy Type Theory HoTT = Martin-L¨ of’s type theory + Voevodsky’s univalence axiom Key ingredients: (1) substitution operation, (2) identity types, (3) Π -types, (4) a type universe, (5) univalence axiom. We give a category-theoretic account of these. 6

󴐼 󴖼 Models of HoTT Definition. A model of homotopy type theory consists of ◮ a category E with a terminal object 1, ◮ a class Fib of maps, called fibrations , subject to axioms (1)-(5). Idea: the sequent x : A ⊢ B ( x ) : type is interpreted as a fibration B p A Warning: issues of coherence will be ignored. 7

󴑜 󴗜 󴐼 󴖼 󴐼 󴖼 Models of HoTT: substitution (1) Pullbacks of fibrations exist and are again fibrations. So for every map σ : A ′ → A we have σ ∗ 󴑜 󴗜 Fib / A ′ Fib / A Diagrammatically: B ′ B p ′ p 󴑜 󴗜 A A ′ σ 8

󴑗 󴗗 󴐼 󴖼 Models of HoTT: identity types (2) For every fibration p : B → A , there is a factorization r 󴑜 󴗜 Id B B q ∆ p B × A B where r ∈ ⋔ Fib and q ∈ Fib. 9

󴑌 󴗌 󴑜 󴗜 󴑋 󴗋 Models of HoTT: Π -types (3) If p : B → A is a fibration, pullback along p has a right adjoint p ∗ 󴑜 󴗜 Fib / A Fib / B We call this the pushforward along p . Note. For a fibration q : C → B , global elements of p ∗ ( q ) are sections of q : 󴑜 󴗜 p ∗ ( C ) A B C ❃ ❄ ❃ ❄ ⑧ 󴒩 󴘩 ③③③③③③③③ ❃ ❄ ⑧ ❃ ❄ ⑧ ❃ ⇔ ❄ ⑧ ❄ ❃ ⑧ ❄ ❃ ⑧ q 1 A 1 B p ∗ ( q ) ❄ ❃ ⑧ 󴒬 󴘬 ⑧ B A 10

󴑜 󴗜 󴐼 󴖼 󴐼 󴖼 Models of HoTT: universes We now assume that there is a notion of ‘smallness’ for the maps of E (e.g. given by a bound on the cardinality of fibers). (4) There is a fibrant object U and a small fibration π : ˜ U → U which weakly classifies small fibrations, i.e. for all such p : B → A there is a pullback ˜ B U ❴✤ p π 󴑜 󴗜 U A Note. We do not ask for uniqueness of the pullback. 11

󴐼 󴖼 󴐼 󴖼 󴑜 󴗜 Models of HoTT: univalence (5) The fibration π : ˜ U → U is univalent. In SSet, this holds if and only if for every small fibration p : B → A , the space of squares 󴑜 󴗜 ˜ B U p π 󴑜 󴗜 U A such that B → A × U ˜ U is a weak equivalence, is contractible. Question: how can we define examples of models of HoTT? 12

Quillen model categories Fix E with a Quillen model structure (Weq , Fib , Cof). Let TrivFib = Weq ∩ Fib , TrivCof = Weq ∩ Cof. Question: Is ( E , Fib) a model of HoTT? Let’s look at the axioms for a model of HoTT: (1) : pullbacks exist and preserve fibrations. (2) : given by factorization as trivial cofibration followed by a fibration. 13

󴒜 󴘜 󴑜 󴗜 The Frobenius property Lemma. Assume that, for a fibration p : B → A , we have p ∗ E / A E / B ⊥ p ∗ TFAE: (i) p ∗ preserves fibrations (ii) p ∗ preserves trivial cofibrations. Definition. A wfs (L , R) is said to have the Frobenius property if pullback along R-maps preserves L-maps. Remark. Assume that Cof = { monomorphisms } . TFAE: (i) The wfs (TrivCof , Fib) has the Frobenius property. (ii) The model structure is right proper, i.e. pullback of weak equivalences along fibrations are weak equivalences. 14

󴑜 󴗜 󴐼 󴖼 󴐼 󴖼 The fibration extension property Assume E = Psh( C ), cofibrations ⊆ monos, fibrations are local (Cisinski). Lemma. Assume π : ˜ U → U classifies small fibrations. Then TFAE: (i) the universe U is fibrant, (ii) small fibrations can be extended along trivial cofibrations, i.e. B ′ B ❴✤ p p ′ 󴑜 󴗜 A ′ A i (iii) small fibrations can be extended along generating trivial cofibrations. We call (ii) the fibration extension property . 15

󴑔 󴗔 󴑄 󴗄 󴑔 󴗔 󴑄 󴗄 Quillen model categories: the glueing property Lemma. TFAE: (i) the fibration π : ˜ U → U is univalent, (ii) weak equivalences between small fibrations can be extended along cofibrations: 󴑜 󴗜 • B 1 ❖ ❖ ✴ ❖ ❖ ✴ ❖ ✴ ❖ w ❖ ✴ ❖ ❖ ✴ ❖ ❖ ✴ ❖ ❖ ✴ ❖ ✴ 󴑜 󴗜 B ′ B 2 ✴ p 1 ✴ 2 ✴ ⑧ ⑦ ⑧ ✴ ⑦ ⑧ ✴ ⑦ q 2 ⑧ ⑦ ✴ ⑧ ⑦ p 2 ⑧ ✴ ⑦ ⑧ 󴒫 󴘫 ⑦ 󴒬 󴘬 ⑧ 󴑜 󴗜 A ′ A i We call (ii) the glueing property (cf. Coquand et al.) 16

Example: simplicial sets Let SSet be the category of simplicial sets. We consider the model structure for Kan complexes. Right properness ◮ via geometric realization (see Hovey, Hirschhorn) ◮ via minimal fibrations (Joyal and Tierney) Fibration extension property ◮ via minimal fibrations and theory of bundles (Joyal) Glueing property ◮ Direct proof (Voevodsky) ◮ Via theory of fiber bundles (Moerdijk) 17

Issues Theorem (Bezem, Coquand, Parmann) . The right properness of SSet cannot be proved constructively. A constructive proof is essential for applications in mathematical logic. How can we fix this? Coquand’s approach ◮ Switch from simplicial sets to cubical sets ◮ Work with uniform fibrations. This is useful also to deal with coherence (Swan, Larrea-Schiavon). Plan: ◮ alternative presentation of cubical set model ◮ analysis via the notions of an algebraic weak factorization system. 18

Goal For a category E , we want: (1) to construct an algebraic weak factorization system (Cof , TrivFib) (2) to construct an algebraic weak factorization system (TrivCof , Fib) (3) to show that (TrivCof , Fib) has the Frobenius property. (4) to prove the glueing property. (5) to prove the fibration extension property (6) to show that we have an algebraic model structure. (1)-(3) this talk, (4)-(6) next talk. The approach to (1)-(2) is inspired by Cisinski’s theory. 19

Part II: Uniform fibrations and the Frobenius property 20

󴑜 󴗜 󴑬 󴗬 󴑌 󴗌 Algebraic weak factorization systems For a weak factorization system, we often ask for ◮ functorial factorizations , i.e. functors ( L , R ) such that f A B ❄ ❄ ⑧ ❄ ⑧ ❄ ⑧ ❄ ⑧ ❄ ⑧ ❄ ⑧ L f R f ❄ ⑧ ⑧ • gives the required factorization. In an algebraic weak factorization system , we also ask that ◮ L has the structure of a comonad, ◮ R has the structure of a monad, ◮ a distributive law between L and R . Grandis and Tholen (2006), Garner (2009). 21

󴑤 󴗤 󴐼 󴖼 󴐼 󴖼 󴑜 󴗜 󴐼 󴖼 󴑠 󴗠 󴑜 󴗜 󴑜 󴗜 󴑤 󴗤 󴑜 󴗜 󴐼 󴖼 󴑜 󴗜 󴑜 󴗜 󴐼 󴖼 Uniform liftings Fix a category E . Let u : I → E → be a functor. Definition. A right I -map is a map p : B → A in E equipped with ◮ a function φ which assigns a diagonal filler s X i B u i p Y i A t for i ∈ I , subject to a uniformity condition: s X j X i B u j p Y j Y i A t I ⋔ = category of right I -maps. 22

The setting (I) Let E be a presheaf category. We assume a functorial cylinder X 󰄴→ I ⊗ X with endpoint inclusions δ k ⊗ X : X → I ⊗ X such that (C1) the cylinder has contractions, ε X : I ⊗ X → X (C2) the cylinder has connections, c k X : I ⊗ I ⊗ X → I ⊗ X (C3) I ⊗ ( − ) has a right adjoint (C4) I ⊗ ( − ) : E → E preserves pullback squares (C5) the endpoint inclusions δ k ⊗ X : X → I ⊗ X are cartesian. Examples: SSet , CSet. 23

The setting (II) We also fix a full subcategory → E → M ↩ cart of monomorphisms such that: (M1) the unique map ⊥ X : 0 → X is in M , for every X ∈ E (M2) M is closed under pullbacks (M3) M is closed under pushout product with the endpoint inclusions. Examples: M = all monomorphisms, in SSet or CSet. 24

󴑜 󴗜 󴐼 󴖼 󴑤 󴗤 󴑜 󴗜 Uniform trivial fibrations Fix E , I ⊗ ( − ), M as above. Write u : M → E → for the inclusion. Definition. A uniform trivial fibration is a right M -map, i.e. a map f : B → A together with a function which assigns fillers s X B i 󴐼 󴖼 f Y A t where i : X → Y is a monomorphism in M , subject to uniformity. TrivFib = M ⋔ = category of uniform trivial fibrations. 25

󴑄 󴗄 󴐼 󴖼 󴑜 󴗜 󴑜 󴗜 󴑔 󴗔 󴐼 󴖼 󴑜 󴗜 Cylinder inclusions For a monomorphism i : X → Y in M , we have the pushout product i X Y δ k ⊗ X δ k ⊗ Y I ⊗ X • δ k ˆ ⊗ i I ⊗ Y I ⊗ i We get a subcategory Cyl ⊆ E → with objects the “cylinder inclusions” δ k ˆ ⊗ i : ( I ⊗ X ) ∪ ( { k } ⊗ Y ) → I ⊗ Y 󰂋 󰂊󰂉 󰂌 • 26